A fundamental question in population ethics is the tradeoff between quantity and quality. The world has finite resources, so if we promote policies that increase the population, we do so at the risk of decreasing quality of life.

Derek Parfit is credited with popularizing the importance of this problem when he pointed out that any population ethic which obeys some seemingly reasonable constraints must end up with what he called "the repugnant conclusion" - the conclusion that a world full of miserable people is better than a sparsely-populated world full of happy people. Since Parfit, there have been a range of theories seeking to preserve our intuitions about ethics while still avoiding this conclusion.

One discovery of abstract algebra is that we can understand the limitations of systems based solely on the questions they are able to answer, even if we don't know what the answers are.

Here, I'll consider any system capable of answering a question like "Are two people who each live 50 years morally equivalent to one person who lives 100 years?" (Again, we don't require that the answer be "Yes" or "No", but merely that there be

*some*answer.) For notational ease, I use the symbol $\oplus$ to be the "moral combination", e.g. the above question can be written $$(50\text{ years})\oplus(50\text{ years})=100\text{ years?}$$ Such a system I will call a "moral group" and require that it obey a few standard requirements. These are:

- Any two people can be replaced with one who is (significantly) better off
- There is some level of welfare which is "morally neutral", i.e. a person of that welfare neither increases nor decreases the overall moral desirability of the world.
- For any level of welfare, no matter how high, there is some level of welfare which is so negative that the two cancel out

With this definition, we have an impossibility theorem:

**Theorem**: In any "moral group", the repugnant conclusion holds.

**Proof**: Suppose that $x$ is a welfare level that is better than "barely worth living". Formally, say that there must be some $y$ where $0 < y < x$, i.e. it's possible to be worse off than $x$ and still have a "life worth living". We'll show that a world with just $x$ is morally equivalent to a world with two people who are both worse off than $x$. Repeating this ad infinitum leads to the conclusion that a world with a few happy people is equivalent to a world with a large number of people whose lives are "barely worth living."

Choose some $y$ between $0$ and $x$ (one exists, by the definition of $x$). Note that $x=y\oplus z$ where $z=y^{-1}\oplus x$, so we just need to show that $z<x$. Since $y>0$, $y^{-1} < 0$ because if it weren't then we'd have $y^{-1} > 0$; adding $y$ to both sides results in $0>y$ which contradicts the assumption that $y>0$. Therefore $y^{-1} \oplus x < x$, or to write it another way: $z < x$. So $x=y\oplus z$, with $y$ and $z$ both worse than $x$.

This means that for any world with people $x_1,x_2,\dots$ of high welfare, there is an equivalent world $y_1,y_2,\dots$ with more people, each of whom have lower welfare. By adding some person of low (but still positive) welfare $y_{n+1}$ to the second world, it becomes better than the first, resulting in the repugnant conclusion.∎

This means that for any world with people $x_1,x_2,\dots$ of high welfare, there is an equivalent world $y_1,y_2,\dots$ with more people, each of whom have lower welfare. By adding some person of low (but still positive) welfare $y_{n+1}$ to the second world, it becomes better than the first, resulting in the repugnant conclusion.∎

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